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area inequality

Source: flanders '97

September 28, 2005
geometryinequalitiestrigonometryratiolimitgeometric sequence

Problem Statement

Δoa1b1\Delta oa_1b_1 is isosceles with ∠a1ob1=36∘\angle a_1ob_1 = 36^\circ. Construct a2,b2,a3,b3,...a_2,b_2,a_3,b_3,... as below, with ∣oai+1∣=∣aibi∣|oa_{i+1}| = |a_ib_i| and ∠aiobi=36∘\angle a_iob_i = 36^\circ, Call the summed area of the first kk triangles AkA_k. Let SS be the area of the isocseles triangle, drawn in - - -, with top angle 108∘108^\circ and ∣oc∣=∣od∣=∣oa1∣|oc|=|od|=|oa_1|, going through the points b2b_2 and a2a_2 as shown on the picture. (yes, cdcd is parallel to a1b1a_1b_1 there) Show Ak<SA_k < S for every positive integer kk. http://www.mathlinks.ro/Forum/album_pic.php?pic_id=284
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